No public trails in the hill country-its just not a thing. Even public access camping spots that aren't a state park are almost non-existent. Honestly not a thing in Texas in general. Be very careful especially in the hill country, all land is private land, and esp if you're a little ways out on a county road, people are a mite suspicious of random strangers. NEVER drive off the actual public road when going through the hill country. You are almost certainly going to be trespassing, and that's not something you want to be doing.
Your best bet to do some actual wheeling and camping is going to an offroad park with camping. Hidden Falls is pretty popular, I've never been there though. Wolf Caves near Mason is great, went there with some buddies a month ago. But Wolf Caves is actual wheeling-the white trail just going around the perimeter of the park offered some fun small challenges-and we all bashed our skids pretty good throughout the day. I don't have a ton of 4x4 wheeling experience, and don't have a ton of armor or a winch, I wouldn't go there without a second vehicle and a spotter who knew more than I did. Not sure how that applies to you, that's just me. If the tundra in your profile is your truck, its prob a bit too big for a lot of the Wolf Caves trails.
Sounds like you're more wanting to do camping/sightseeing, when me and my wife want to do that we just pick a direction and let google maps take us there "avoiding highways". If you're driving through the hill country, it will be interesting.
Also, keep in mind online reservations for state parks in central Texas book up months in advance, especially after COVID had them shut down for a while. If you have a RTT or trailer you'll basically be limited to RV or spots with at least electrical. If you've got a regular tent you can do the walk in spots, but you'll be possibly hauling your stuff a few hundred yards, or even a 1/4 mile in a couple of cases.
Hill Country Natural Area near Tarpley is nice and removed from the beaten path, some creek crossings on some of the county roads in that area, and Tarpley has a great burger joint. It won't feel remote like wilderness, but its definitely got that quiet, small Texas town feel to it.
There's a hill country sightseeing route posted somewhere on Overland Bound and Expedition portal, all the same thing. If you feel like playing with GAIA or your offroad navigation app of choice, importing/exporting map files etc, its a nice day plus long drive, and really quiet roads. Everything except the offroad parks mentioned are all maintained dirt county roads, with some creek crossings in the mix. Pretty tame, but also still very removed from things like gas stations and food/water. Not too far driving, but if you were to break down on one of those roads it would suck. No need for extra fuel or anything, just gas up before you get way off.
